Triangulation How-To

Ok, so now we know how to measure angles and draw the corresponding lines on our pad. So how do you go about drawing this handsome fellow by triangulation (at last)?

Remember the side-angle-angle theorem? One side and two angles determine a triangle. Meaning they determine the position of the missing vertex once you know two vertices.

So we start by choosing an arbitrary line to be the given side in the side-angle-angle theorem. I like to start from big to small, so I'll try to choose a line joining two distant points in the figure. Also, I like to start simple, so I choose a simple vertical alignement.

I notice by scanning vertically with my pencil that the central point of the head aligns vertically with the inside of the right leg and with the point where the foot intersects the inside of the leg. The most important thing is that this is a line I'll have no problem in finding again. I choose two points on that lign. The first is the point where the ridge of the head (see the highlight there?) intersects that line, and the second is the top of the foot where it intersects both the leg and the line. I cannot forget where those point are, so I make sure where they are and memorize them.

Now what do I draw? I draw on my pad a line making the same angle as this one. In this case the angle is 90 degrees, it is a vertical line. How big do i draw it? As big as I want. The size of the line I draw is arbitrary as long as it has the same angle with the vertical of the pad as the line on the model has with the true vertical. How big I choose to make this line determines only the scale. I choose the line according to the size I want the figure to occupy on my drawing. After I make this initial choice, the scale is fixed.

Now I measure the angle of the line that joins the vertex of the head to the center of the joint of the elbow. On my pad I make a line (a faint line, to erase later) that starts on the top of the first line I made and makes the same angle as I measured on the model.

Now I measure the angle from the point at the foot to the same point on the elbow. I draw the line on my pad, starting at the point of the foot (the low end of the first line I put on the drawing) and making the same angle with the pad as I measured on the model.

Now, the two lines will intersect at some point. That point is the point on my drawing where the elbow is located! So I found the elbow by triangulation, that is, I constructed a triangle, by measuring two angles from a given side, such that the found vertex is the center of the elbow.

Ok, so I don't really need the construction lines anymore, I can erase them. What is left is three points: Head, foot, elbow. From these I can find others by triangulation, once more.

Here I find the other elbow, in the same way as before. Now I have four points.

Time for a checkpoint: up to now I measured everything from my initial two points. Now I check for consistency. I measure the the angle of the line that joins the two elbows on the model. If it checks out with the angle on my drawing I proceed. If not, I go back and check what I did. The first measurements are the hardest and the most important, so make sure of them.

When everything checks out we proceed: We triangulate the shoulder from the two elbows: measure the angle from one elbow to the shoulder, then from the other, intersect the two lines and you have the shoulder point.

Then proceed like this: triangulate the other shoulder, then check for consistency from one shoulder to the other. Also check the triangle that the head makes with the shoulder girdle (the line between the two shoulders).

Then find, in the same way, the main joints: the knees, ankles, the points of the hips. Stick to points you can name and identify readily. The biggest danger is getting confused and not knowing from where you measured what. Especially at the start, it is probably good not to erase construction lines (with practice you won't even make the lines, just the points mostly, but for now go easy)

A good exercise would be to plot all these green points. They are all anatomical landmarks (in the anatomy of wooden dolls :)). Start with a doll if possible, then try it on a human. At first you'll find it hard, but later you'll find that these are far too many points. I use this thing to do quick anatomical drawings, just by tracing the main lines: limb directions, position of joints, no need for measuring lenghts because the lengths come automatically from the angles of the imaginary lines.

Always focus on points you can identify. On humans focus on bony landmarks, try to get a scheme for what you want, so that you won't ever be in doubt in the middle of a drawing over what you measured or from where. Good landmarks are the usual: joints, ridges, places where angles change abruptly. Also, high contrast intersections of soft tissues, but only in special cases as these depend on static positions and any small move breaks them apart. Bones are reliable.

This last one would be really going too far. But it would be easy, from the other, and from this point you would just be painting by numbers. Which is not the point at all, so remember to distinguish between an exercise and a way of drawing. After a while it is up to you to learn what measurements you need. You don't want or need milimetric precision, you want a quick and effective method to find the major measurements. After you have those, be brave and just look, and see, and draw.

Your ultimate purpose is to need no measuring at all.

next: how to calibrate yourself.